第 33831 題：梯度向量

原題與已知條件

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-boundary；明示資料={"mode":"positive_gradient_direction","model":{"gradient":[6,-1]}}；請依 exact_ordered_tuple_declared_by_interface 回答。

• G001 — fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-boundary；明示資料={"mode":"positive_gradient_direction","model":{"gradient":[6,-1]}}；請依 exact_ordered_tuple_declared_by_interface 回答。

定義與適用範圍

沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。

• mko-gradient-vector

逐步推導

source roles = [qf-mod50-graddeep-boundary, [domain, mode, model, operation]]

derived = [graddeep-boundary, Exact_gradient_vector_density]

source roles = [positive direction ORIGINAL gradient raw coordinate0, 100000000, 6, 1]

derived = [[6, 1], 6]

positive direction ORIGINAL gradient raw coordinate0 ORIGINAL numerator over denominator := div(6, 1)

positive direction ORIGINAL gradient raw coordinate0 ORIGINAL numerator over denominator = 6

source roles = [positive direction ORIGINAL gradient raw coordinate1, 100000000, -1, 1]

derived = [[-1, 1], -1]

positive direction ORIGINAL gradient raw coordinate1 ORIGINAL numerator over denominator := div(-1, 1)

positive direction ORIGINAL gradient raw coordinate1 ORIGINAL numerator over denominator = -1

source roles = [positive direction ORIGINAL gradient, [6, -1]]

derived = [[6, -1]]

source roles = [positive gradient inner product product0 SOURCE naive mul, 100000000, 36, 1]

derived = [[36, 1], 36]

positive gradient inner product product0 SOURCE naive mul ORIGINAL numerator over denominator := div(36, 1)

positive gradient inner product product0 SOURCE naive mul ORIGINAL numerator over denominator = 36

positive gradient inner product product0 ORIGINAL rational mul := mul(6, 6)

positive gradient inner product product0 ORIGINAL rational mul = 36

source roles = [positive gradient inner product prefix0 SOURCE naive sum, 100000000, 36, 1]

derived = [[36, 1], 36]

positive gradient inner product prefix0 SOURCE naive sum ORIGINAL numerator over denominator := div(36, 1)

positive gradient inner product prefix0 SOURCE naive sum ORIGINAL numerator over denominator = 36

positive gradient inner product prefix0 ORIGINAL rational sum := sum(0, 36)

positive gradient inner product prefix0 ORIGINAL rational sum = 36

source roles = [positive gradient inner product product1 SOURCE naive mul, 100000000, 1, 1]

derived = [[1, 1], 1]

positive gradient inner product product1 SOURCE naive mul ORIGINAL numerator over denominator := div(1, 1)

positive gradient inner product product1 SOURCE naive mul ORIGINAL numerator over denominator = 1

positive gradient inner product product1 ORIGINAL rational mul := mul(-1, -1)

positive gradient inner product product1 ORIGINAL rational mul = 1

source roles = [positive gradient inner product prefix1 SOURCE naive sum, 100000000, 37, 1]

derived = [[37, 1], 37]

positive gradient inner product prefix1 SOURCE naive sum ORIGINAL numerator over denominator := div(37, 1)

positive gradient inner product prefix1 SOURCE naive sum ORIGINAL numerator over denominator = 37

positive gradient inner product prefix1 ORIGINAL rational sum := sum(36, 1)

positive gradient inner product prefix1 ORIGINAL rational sum = 37

source roles = [positive gradient inner product, [6, -1], [6, -1]]

derived = [37]

source roles = [[6, -1]]

derived = [37, 是, 否]

逐步反向核對

• S020 · premise_and_result_audit — 反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。
• S019 · premise_and_result_audit — 反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。
• S018 · premise_and_result_audit — 以精確有理數重算 positive gradient inner product prefix1 ORIGINAL rational sum，不以小數近似替代。
• S017 · premise_and_result_audit — 以精確有理數重算 positive gradient inner product prefix1 SOURCE naive sum ORIGINAL numerator over denominator，不以小數近似替代。
• S016 · premise_and_result_audit — 反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。
• S015 · premise_and_result_audit — 以精確有理數重算 positive gradient inner product product1 ORIGINAL rational mul，不以小數近似替代。
• S014 · premise_and_result_audit — 以精確有理數重算 positive gradient inner product product1 SOURCE naive mul ORIGINAL numerator over denominator，不以小數近似替代。
• S013 · premise_and_result_audit — 反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。
• S012 · premise_and_result_audit — 以精確有理數重算 positive gradient inner product prefix0 ORIGINAL rational sum，不以小數近似替代。
• S011 · premise_and_result_audit — 以精確有理數重算 positive gradient inner product prefix0 SOURCE naive sum ORIGINAL numerator over denominator，不以小數近似替代。
• S010 · premise_and_result_audit — 反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。
• S009 · premise_and_result_audit — 以精確有理數重算 positive gradient inner product product0 ORIGINAL rational mul，不以小數近似替代。
• S008 · premise_and_result_audit — 以精確有理數重算 positive gradient inner product product0 SOURCE naive mul ORIGINAL numerator over denominator，不以小數近似替代。
• S007 · premise_and_result_audit — 反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。
• S006 · premise_and_result_audit — 反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。
• S005 · premise_and_result_audit — 以精確有理數重算 positive direction ORIGINAL gradient raw coordinate1 ORIGINAL numerator over denominator，不以小數近似替代。
• S004 · premise_and_result_audit — 反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。
• S003 · premise_and_result_audit — 以精確有理數重算 positive direction ORIGINAL gradient raw coordinate0 ORIGINAL numerator over denominator，不以小數近似替代。
• S002 · premise_and_result_audit — 反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。
• S001 · premise_and_result_audit — 反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。

結論

(positive_gradient_direction, [6, -1], [6, -1], 37, 是, 否, positive_gradient_is_first_order_ascent_not_descent)

依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。

本題覆核與來源綁定

覆核日期：2026-10-26

原題、答案、步驟表達式、數值 AST 與既有歷史保持原樣；原來的候選狀態屬於來源歷史，不覆寫為形式化證明。